Pseudocomplemented semilattices are finite-to-finite relatively universal

• Autores: M. E. Adams, Jürg Schmid
• Localización: Algebra universalis, ISSN 0002-5240, Vol. 58, Nº. 3, 2008, págs. 303-333
• Idioma: inglés
• DOI: 10.1007/s00012-008-2071-3
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• Resumen

  • It is shown that the category of directed graphs is isomorphic to a subcategory of the variety S of all pseudocomplemented semilattices which contains all homomorphisms whose images do not lie in the subvariety B of all Boolean pseudocomplemented semilattices. Moreover, the functor exhibiting the isomorphism may be chosen such that each finite directed graph is assigned a finite pseudocomplemented semilattice. That is to say, it is shown that the variety S of all pseudocomplemented semilattices is finite-to-finite B-relatively universal.

    This illustrates the complexity of the endomorphism monoids of pseudocomplemented semilattices since it follows immediately that, for any monoid M, there exists a proper class of non-isomorphic pseudocomplemented semilattices such that, for each member S, the endomorphisms of S which do not have an image contained in the skeleton of S form a submonoid of the endomorphism monoid of S which is isomorphic to M.